Lecture 6Lecture 6

MO Theory of Extended Molecules and Solids

Suggested reading: Chapter 2.12 & 3.19-3.2Reminder: Quiz WednesdayReminder: Quiz Wednesday

Why is MO Theory Important?

) d h h f l l1) Predicting the shapes of molecules2) Understanding the electronic properties of

molecules and solidsmolecules and solids3) Applications to nanoscience

3) MO Theory of Nanostructures

Au shell

SiO2

Halas Group, Rice University

3) MO Theory of Nanostructures

Au shell

SiO2

Halas Group, Rice University

Nanoparticle Application: Optical Cancer Therapy

Atwater, “The Power of Plasmonics,” Scientific American

1. Shapes of Molecules

W ter (H O)Water (H2O)

Beryllium Hydride (BeH2)

Correlation diagram: shows how one set of orbitals evolves into another as a parameter (such as a bond angle) is changed

Walsh Diagram for XH2

SALCs for XH2: Linear Molecule

221 cc sg

423 ccyu p

zyu p ,2

y

SALCs for XH2: Angular Molecule

322211 ccczpsa

xpb 21

5242 ccypb y

Walsh Diagram for XH2

XH2 molecules with four or less l l t ill b livalence electrons will be linear

XH2 with five to eight valence electrons is predicted to be angular

BeH2 BH2 CH2 NH2 OH2

180o 131o 136o 103o 105o

Hybridization

Molecular shapes can also be intuited by hybridization - a Molecular shapes can also be intuited by hybridization a mixing of atomic orbitals (predates MO theory, but still is conceptually useful)

Si Hybridization

Isolated Si atom Si just before bonding

An isolated Si atom has two electrons in the 3s and two electrons in the 3p orbitals.When Si is about to bond, the one 3s orbital and the three 3p orbitals interfere to form four

hybridized orbitals, hyb, called sp3 orbitals, which are directed toward the

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw‐Hill, 2005)

y hyb pcorners of a tetrahedron. The hyb orbital has a large major lobe and a small back lobe. Each

hyb orbital takes one of the four valence electrons.

Types of Hybridization

sp3 hybridization: Hybridized orbitals formed from an “s” orbital and 3 “p” orbitals.

sp2 hybridization: Hybridized orbitals formed from an “s” orbital and 2 “p” orbitals. Bond angles of 120o

sp hybridization: Hybridized orbitals formed from an “s” orbital and 1 “p” orbitals. Bond angles of 180oand 1 p orbitals. Bond angles of 180

Geometries of Hybrid Orbitals

Note: Hybridization incorrectly predicts the photoelectron spectra of many molecules Note: Hybridization incorrectly predicts the photoelectron spectra of many molecules. Also, hybridization over-emphasizes the localization of bonding electrons. It should

only be used for a qualitative understanding of molecular structure.

2. Electronic Properties of Molecules & Solids

0 154 a=0.154 nm

a=0.142 nm

Graphene: Top 10 list10) The largest aromatic 10) The largest aromatic molecule: One atom thick planar sheet of sp2 bonded carbon9) First isolated in 2004 using scotch tape8) Linear dispersion: completely free electrons8) Linear dispersion: completely free electrons7) Resistivity of 10-6Ohm-cm, less than that of Ag: the lowest resistivity at room temperaturey p6) Absorbs πα~2.3% of white light5) Thermal conductivity exceeding diamond4) Breaking strength 200x steel strongest material3) Large surface area to volume ratio ideal sensor2) St t li d fi ld ( d l t FET )2) Strong response to applied fields (modulators, FETs)1) Zero effective mass electrons and holes

Three-atom system: three energy levels

The energies of the three molecular orbitals, labeled a, b, and c, in a system with three H atoms.

N-atom system: N energy levels

The formation of 2s energy band from the 2s orbitals when N Li atoms (1s2 2s1) come together The formation of 2s energy band from the 2s orbitals when N Li atoms (1s 2s ) come together to form the Li solid. There are N 2s electrons, but 2N states in the band. The 2s band is

therefore only half full. The atomic 1s orbital is close to the Li nucleus and remains undisturbed in the solid.

Band theory of solids

As Li atoms are brought together from infinity, the atomic orbitals overlap and give rise to bands. Outer orbitals overlap first. The 3s orbitals give rise to the 3s band, 2p orbitals to the 2p

band, etc. The various bands overlap to produce a single band in which the energy is nearly p p g gy ycontinuous.

Note: As with MO theory of molecules, we can no longer consider the electrons as belonging to specific atoms – they are shared among the entire solid.

Band theory of solids

In a metal the various energy bands overlap to give a single energy band In a metal, the various energy bands overlap to give a single energy band that is only partially full of electrons. There are states with energies up to

the vacuum level, where the electron is free.

Band theory of solids: The Fermi Level

T i l l t b d di f t l All th l l t Typical electron energy band diagram for a metal. All the valence electrons are in an energy band, which they only partially fill. The top of the band is

the vacuum level, where the electron is free from the solid (PE = 0).

How are the energies of electrons distributed in a band?

1 state with the highest energy highest energy

Many states with comparable

intermediate intermediate energies (given by

the # of nodes)

h h1 state with the lowest energy

Density of States, g(E)

g(E) is the number of states (molecular orbitals) in the energy interval E to (E+dE) per unit volume of the sample (E+dE) per unit volume of the sample.

The total number of states per unit volume up to some energy E’ is:

'

0

)()(E

V dEEgES

Let’s consider the electrons in the solid to be in a 3D potential well of

Deriving the density of states in a metal

Let s consider the electrons in the solid to be in a 3D potential well of volume V, with sides L. The energy of the electrons is:

2222

nnnhE

In 2D, we only have n1 and n2. Each state or electron wavefunctions in the crystal can be represented by a box at n n

32128321nnn

mLE nnn

Each state, or electron wavefunctions in the crystal, can be represented by a box at n1, n2.

How many combinations of n1 and n h e n energ less n2 have an energy less than E’?

Let’s consider the electrons in the solid to be in a 3D potential well of

Deriving the density of states in a metal

Let s consider the electrons in the solid to be in a 3D potential well of volume V, with sides L. The energy of the electrons is:

2222

nnnhE

In 2D, we only have n1 and n2. Each state or electron wavefunctions in the crystal can be represented by a box at n n

32128321nnn

mLE nnn

Each state, or electron wavefunctions in the crystal, can be represented by a box at n1, n2.

The area contained by n1

and n2 approximates a circle with area ¼(πn’2)

In three dimensions, the volume defined by a sphere of radius n' and the

Deriving the density of states in a metal

positive axes n1, n2, and n3, contains all the possible combinations of positive n1, n2, and n3 values that satisfy 22

322

21 nnnn

1 3'61 nVsphere

1

Each orbital can hold two electrons, so the total number of states up to some quantum number n’ is:

3'312)( nVnS sphere

1

Deriving the density of states in a metal

3'312)( nVnS sphere

2

2223

22

212

2

88321 mLnhnnn

mLhE nnn 2

'22 8'

hEmLn

3

2/33

3)'8()'(

hmELES

3h

Since L3 is the volume of the solid, the total number of states per unit volume is:

2/3)'8( E

2/3mdS

3

2/3

3)'8()'(

hmEESV

2/12

2/128)( Ehm

dEdSE eV

g

Applying the Density of States: STM Image of an elliptical atomic corral (Co atoms on a Cu surface)

From Manorahan et al. Nature, Feb. 3, 2000

The Quantum Mirage: An application of the DOS

aphy

Scanning tunneling

Topo

grag g

microscope tip

    states)

Co

dI/dV 

density

 of s

Cu (111)

(d

Mirage

Manoharan, H. C.; Lutz, C. P.; Eigler, D. M. Nature 403 (2000)

The Quantum Mirage: An application of the DOS

aphy

Scanning tunneling

Topo

grag g

microscope tip

    states)

Co

dI/dV 

density

 of s

Cu (111)

(d

Mirage

Manoharan, H. C.; Lutz, C. P.; Eigler, D. M. Nature 403 (2000)

Photonic Crystals: An application of the optical density of statesy

http://photonics.tfp.uni‐karlsruhe.de/research.html

From Wikipedia: “Some structures can completely inhibit the propagation of light of certain colors (energies), creating a photonic bandgap: the DOS is zero for those photon energies.”

Application the DOS: Fermi EnergyAt T=0, the Fermi energy is the chemical potential of the system. For , gy p ya metal, we can readily determine the chemical potential by integrating the density of states:

3

2/3

0 3)8()()(

hmEdEEgE F

E

V

F S

)21('

312

33

VNqnVS sphere

b f l dN: Number of atoms in solidq: Number of electrons contributed per atom

V: volume of solid½: Each state can hold two electrons

3/22 3Nqh 3/2)3(

8 VNq

mhE

eF

Energy bands arise from bonding & antibondingwavefunctions

Left: potential energy versis atomic separation. At the equilibrium distance for Si, a bandgap is formed.

Right: Simple energy band diagram of a semiconductor. CB is the conduction band and VB is the valence band. AT 0 K, the VB is full with all the valence

electrons.

Extra Slides: 1 S d B d P i1. Structure and Bond Properties

2. Determining Oxidation Numbers3 S3. Symmetry

Structure and Bond Properties

Covalent radius: equilibrium bond lengths can be empirically and approximately partitioned into contributions from each atom of the bonded pairthe bonded pair

Bond dissociation enthalpy: the standard reaction enthalpy for the reaction: AB(g) A(g)+B(g)

Pauling electronegativity can be determined from mean A-B bond enthaplies. Binary compounds with electronegativity differences greater than 1.7 tend to be ionic.

Oxidation states are determined by exaggerating the ionic character of a bond. (Charge an atom would have if it acquired the two electrons of a bond )the two electrons of a bond.)

Determining Oxidation Number

Symmetry operations